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Subtracting an integer is equivalent to adding its opposite: a − b = a + (−b). This single rule converts every subtraction problem into an addition problem, which students already know how to handle. For example, 3 − (−5) = 3 + 5 = 8, and −2 − 4 = −2 + (−4) = −6. The "add the opposite" rule is not just a trick — it reflects the deep algebraic structure that subtraction is not a separate operation but addition with an inverse. This concept is essential for simplifying expressions, solving equations, and working with polynomials.
Demonstrate on the number line: subtracting a positive means moving left, subtracting a negative means moving right (reversing direction). Use integer chip models to show that removing a negative chip is the same as adding a positive one. Once students see why the rule works, practice converting subtraction to addition before computing. Emphasize double negatives: −(−5) = +5.
From adding integers, you know how to handle sums like (−3) + (−5) = −8 and 7 + (−4) = 3. Subtraction of integers builds directly on this: the core rule is that subtracting a number is the same as adding its opposite. Written algebraically: a − b = a + (−b). This is not a trick or a shortcut — it is a definition. Subtraction is literally just addition with a sign flip on the second number. Once you apply this rule, every subtraction problem becomes an addition problem you already know how to solve.
Try it on a few examples. 9 − 4 becomes 9 + (−4) = 5, which matches the ordinary arithmetic you already know. Now the interesting cases: 3 − (−5) becomes 3 + (+5) = 8. Subtracting a negative flips it to a positive — you move right on the number line, not left. And −2 − 7 becomes −2 + (−7) = −9. On the number line, subtracting a positive number means moving left: you start at −2 and move 7 units left to reach −9.
The number line makes this geometric. Moving right corresponds to adding a positive; moving left corresponds to subtracting a positive (or equivalently, adding a negative). Here is the key insight for double negatives: subtracting a negative means reversing the left-going direction — so you move right instead. Subtracting −5 is the same as moving 5 units to the right, just like adding +5. Integer chip models say the same thing differently: removing a negative chip from a pile has the same net effect as adding a positive chip. Whether you think about it geometrically or algebraically, the conclusion is identical: two negatives in a subtraction produce a positive effect.
The procedure to follow every time: (1) rewrite the subtraction as addition of the opposite, (2) then apply your integer addition rules. Never try to "compute" a subtraction involving negatives without converting first — this is where errors creep in. For example, (−6) − (−2): convert to (−6) + 2, then apply the rule for adding integers with different signs: |6| − |2| = 4, keep the sign of the larger absolute value (negative), result = −4. Step 1 (convert) and Step 2 (add) are two separate, clean operations that together handle any integer subtraction correctly.